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ThermodynamicsHenri J.F.JansenDepartment of PhysicsOregon State UniversityAugust 19,2010IIContentsPART I.Thermodynamics Fundamentals 11 Basic Thermodynamics.31.1 Introduction.............................41.2 Some deﬁnitions...........................71.3 Zeroth Law of Thermodynamics..................121.4 First law:Energy...........................131.5 Second law:Entropy.........................181.6 Third law of thermodynamics.....................311.7 Ideal gas and temperature......................321.8 Extra questions............................361.9 Problems for chapter 1........................382 Thermodynamic potentials and 492.1 Internal energy............................512.2 Free energies..............................572.3 Euler and Gibbs-Duhem relations..................612.4 Maxwell relations...........................642.5 Response functions...........................652.6 Relations between partial derivatives.................682.7 Conditions for equilibrium......................722.8 Stability requirements on other free energies............782.9 A magnetic puzzle...........................802.10 Role of ﬂuctuations..........................852.11 Extra questions............................942.12 Problems for chapter 2........................963 Phase transitions.1073.1 Phase diagrams...........................1083.2 Clausius-Clapeyron relation......................1163.3 Multi-phase boundaries........................1213.4 Binary phase diagram.........................1233.5 Van der Waals equation of state...................1263.6 Spinodal decomposition........................138IIIIV CONTENTS3.7 Generalizations of the van der Waals equation...........1423.8 Extra questions............................1433.9 Problems for chapter 3........................144PART II.Thermodynamics Advanced Topics 1524 Landau-Ginzburg theory.1534.1 Introduction..............................1544.2 Order parameters...........................1564.3 Landau theory of phase transitions..................1624.4 Case one:a second order phase transition..............1644.5 Order of a phase transition......................1694.6 Second case:ﬁrst order transition..................1704.7 Fluctuations and Ginzburg-Landau theory.............1764.8 Extra questions............................1834.9 Problems for chapter 4........................1845 Critical exponents.1935.1 Introduction..............................1945.2 Mean ﬁeld theory...........................2005.3 Model free energy near a critical point................2085.4 Consequences of scaling........................2115.5 Scaling of the pair correlation function...............2175.6 Hyper-scaling..............................2185.7 Validity of Ginzburg-Landau theory.................2195.8 Scaling of transport properties....................2215.9 Extra questions............................2255.10 Problems for chapter 5........................2266 Transport in Thermodynamics.2296.1 Introduction..............................2306.2 Some thermo-electric phenomena...................2336.3 Non-equilibrium Thermodynamics..................2376.4 Transport equations..........................2406.5 Macroscopic versus microscopic....................2456.6 Thermo-electric eﬀects........................2526.7 Extra questions............................2626.8 Problems for chapter 6........................2627 Correlation Functions.2657.1 Description of correlated ﬂuctuations................2657.2 Mathematical functions for correlations...............2707.3 Energy considerations.........................275PART III.Additional Material 279CONTENTS VA Questions submitted by students.281A.1 Questions for chapter 1.......................281A.2 Questions for chapter 2.......................284A.3 Questions for chapter 3.......................287A.4 Questions for chapter 4.......................290A.5 Questions for chapter 5.......................292B Summaries submitted by students.295B.1 Summaries for chapter 1......................295B.2 Summaries for chapter 2......................297B.3 Summaries for chapter 3......................299B.4 Summaries for chapter 4......................300B.5 Summaries for chapter 5......................302C Solutions to selected problems.305C.1 Solutions for chapter 1.......................305C.2 Solutions for chapter 2.......................321C.3 Solutions for chapter 3.......................343C.4 Solutions for chapter 4.......................353VI CONTENTSList of Figures1.1 Carnot cycle in PV diagram...................201.2 Schematics of a Carnot engine..................211.3 Two engines feeding eachother..................221.4 Two Carnot engines in series...................252.1 Container with piston as internal divider............532.2 Container where the internal degree of freedom becomes ex-ternal and hence can do work...................543.1 Model phase diagram for a simple model system........1113.2 Phase diagram for solid Ce.....................1123.3 Gibbs energy across the phase boundary at constant temper-ature,wrong picture.........................1133.4 Gibbs energy across the phase boundary at constant temper-ature,correct picture........................1143.5 Volume versus pressure at the phase transition.........1143.6 Model phase diagram for a simple model system in V-T space.1153.7 Model phase diagram for a simple model system in p-V space.1163.8 Gibbs energy across the phase boundary at constant temper-ature for both phases........................1183.9 Typical binary phase diagramwith regions L=liquid,A(B)=Bdissolved in A,and B(A)=A dissolved in B...........1243.10 Solidication in a reversible process...............1253.11 Typical binary phase diagram with intermediate compoundAB,with regions L=liquid,A(B)=B dissolved in A,B(A)=Adissolved in B,and AB= AB with either A or B dissolved...1263.12 Typical binary phase diagram with intermediate compoundAB,where the intermediate region is too small to discernfrom a line...............................1273.13 Impossible binary phase diagramwith intermediate compoundAB,where the intermediate region is too small to discernfrom a line...............................1273.14 Graphical solution of equation 3.24................1313.15 p-V curves in the van der Waals model..............133VIIVIII LIST OF FIGURES3.16 p-V curves in the van der Waals model with negative valuesof the pressure............................1333.17 p-V curve in the van der Waals model with areas correspond-ing to energies............................1353.18 Unstable and meta-stable regions in the van der Waals p-Vdiagram................................1383.19 Energy versus volume showing that decomposition lowers theenergy.................................1414.1 Heat capacity across the phase transition in the van der Waalsmodel..................................1554.2 Heat capacity across the phase transition in an experiment..1554.3 Continuity of phase transition around critical point in p-Tplane..................................1584.4 Continuity of phase around singular point............1594.5 Continuity of phase transition around critical point in H-Tplane..................................1604.6 Magnetization versus temperature.................1654.7 Forms of the Helmholtz free energy................1664.8 Entropy near the critical temperature..............1674.9 Specic heat near the critical temperature............1684.10 Magnetic susceptibility near the critical temperature.....1694.11 Three possible forms of the Helmholtz free energy in case 2.1714.12 Values for m corresponding to a minimum in the free energy.1724.13 Magnetization as a function of temperature...........1734.14 Hysteresis loop............................1744.15 Critical behavior in rst order phase transition.........1766.1 Essential geometry of a thermocouple...............234C.1 m versus T-H............................363C.2 Figure 1................................366INTRODUCTION IXIntroduction.Thermodynamics???Why?What?How?When?Where?Many questionsto ask,so we will start with the ﬁrst one.A frequent argument against studyingthermodynamics is that we do not have to do this,since everything follows fromstatistical mechanics.In principle,this is,of course,true.The argument,how-ever,assumes that we know the exact description of a systemon the microscopicscale,and that we can calculate the partition function.In practice,we can onlycalculate the partition function for a few simple cases,and in all other cases weneed to make serious approximations.This is where thermodynamics plays aninvaluable role.In thermodynamics we derive basic equations that all systemshave to obey,and we derive these equations from a few basic principles.Inthis sense thermodynamics is a meta-theory,a theory of theories,very similarto what we see in a study of non-linear dynamics.Thermodynamics gives usa framework for the results derived in statistical mechanics,and allows us tocheck if approximations made in statistical mechanical calculations violate someof these basic results.For example,if the calculated heat capacity in statisticalmechanics is negative,we know we have a problem!There are some semantic issues with the words thermodynamics and sta-tistical mechanics.In the English speaking part of the world thermodynamicsis often seen as a subset of the ﬁeld of statistical mechanics.In the Germanworld it is often seen as an a diﬀerent ﬁeld from statistical mechanics.I takethe latter view.Thermodynamics is the ﬁeld of physics describing thermal ef-fects in matter in a manner which is independent of the microscopic details ofthe material.Statistical mechanics starts at a microscopic model and derivesconclusions for the macroscopic world,based on these microscopic details.Inthis course we discuss thermodynamics,we present equations and conclusionswhich are independent of the microscopic details.Thermodynamics also gives us a language for the description of experimen-tal results.It deﬁnes observable quantities,especially in the form of responsefunctions.It gives the deﬁnitions of critical exponents and transport properties.It allows analyzing experimental data in the framework of simple models,likeequations of state.It provides a framework to organize experimental data.Tosay that we do not need this is quite arrogant,and assumes that if you can-not follow the (often very complicated) derivations in statistical mechanics youmight as well give up.Thermodynamics is the meeting ground of experimentersand theorists.It gives the common language needed to connect experimentaldata and theoretical results.Classical mechanics has its limits of validity,and we need relativity and/orquantum mechanics to extend the domain of this theory.Thermodynamics andstatistical mechanics do not have such a relation,though,contrary to what peo-ple claim who believe that we do not need thermodynamics.A prime exampleis the concept of entropy.Entropy is deﬁned as a measurable quantity in ther-X INTRODUCTIONmodynamics,and the deﬁnition relies both on the thermodynamic limit (a largesystem) and the existence of reservoirs (an even larger outside).We can alsodeﬁne entropy in statistical mechanics,but purists will only call this an entropyanalogue.It is a good one,though,and it reproduces many of the well knownresults.The statistical mechanical deﬁnition of entropy can also be applied tovery small systems,and to the whole universe.But problems arise if we nowalso want to apply the second law of thermodynamics in these cases.Smallsystem obey equations which are symmetric under time reversal,which contra-dicts the second law.Watch out for Maxwell’s demons!On the large scale,theentropy of the universe is probably increasing (it is a very large system,andby deﬁnition isolated).But if the universe is in a well deﬁned quantum state,the entropy is and remains zero!These are very interesting questions,but fora diﬀerent course.Confusing paradoxes arise easily if one does not appreciatethat thermodynamics is really a meta-theory,and when one applies conceptsunder wrong conditions.Another interesting question is the following.Do large systems obey thesame equations as small systems?Are there some new ingredients we need whenwe describe a large system?Can we simply scale up the microscopic models toarrive at the large scale,as is done in renormalization group theory?How doesthe arrow of time creep into the description when we go from the microscopictime reversible world to the macroscopic second law of thermodynamics?Howdo the large scale phenomena emerge from a microscopic description,and whydo microscopic details become unimportant or remain observable?All goodquestions,but also for another course.Here we simply look at thermodynamics.And what if you disagree with what was said above?Keep reading never-theless,because thermodynamics is also fun.Well,at least for me it is......The material in front of you is not a textbook,nor is it an attempt at afuture textbook.There are many excellent textbooks on thermodynamics,andit is not very useful to add a new textbook of lower quality.Also,when youwrite a textbook you have to dot all the t-s and cross all the i-s,or somethinglike that.You get it,I am too lazy for that.This set of notes is meant to bea tool to help you study the topic of thermodynamics.I have over the yearscollected the topics I found relevant,and working through these notes will giveyou a good basic understanding of what is needed in general.If any importanttopic is missing,I would like to know so I can add it.If you ﬁnd a topic too farout,so be it.All mistakes in these notes are mine.If something is quite useful,it is stolen from somewhere else.You can simply take these notes and read them.After doing so,you willat least have seen the basic concepts,and be able to recognize them in theliterature.But a much better approach is to read these notes and use them as astart for further study.This could mean going to the library and looking up thesetopics in a number of books on thermodynamics.Nothing helps understandingmore than seeing diﬀerent descriptions of the same material.If there is one skillthat is currently missing among many students,it is the capability of reallyusing a library!Also,I do not want to give you examples of what I considergood textbooks.You should go ﬁnd out.My opinion would only be a singleINTRODUCTION XIbiased opinion anyhow.These notes started when I summarized discussions in class.In the currentform,I have presented them as reading material,to start class discussions.Thermodynamics can be taught easily in a non-lecture approach,and I amworking on including more questions which could be discussed in class (they areespecially lacking in the later parts).Although students feel uneasy with thisapproach,having a fear that they miss something important,they should realizethat the purpose of these notes is to make sure that all important material isin front of them.Class discussions,of course,have to be guided.Sometimes adiscussion goes in the wrong direction.This is ﬁne for a while,but than theinstructor should help bring it back to the correct path.Of course,the analysisof why the discussion took a wrong turn is extremely valuable,because onelearns most often from one’s mistakes (at least,one should).To be honest,ﬁnding the right balance for each new group remains a challenge.The material in these notes is suﬃcient for a quarter or a semester course.In a semester course one simply adds expansions to selected topics.Also,thematerial should be accessible for seniors and ﬁrst year graduate students.Themathematics involved is not too hard,but calculus with many partial derivativesis always a bit confusing for everybody,and functional derivatives also need a bitof review.It is assumed that basic material covered in the introductory physicssequence is known,hence students should have some idea about temperatureand entropy.Apart from that,visit the library and discover some lower leveltexts on thermodynamics.Again,there are many good ones.And,if thesetextbooks are more than ten years old,do not discard them,because they arestill as relevant as they were before.On the other hand,if you use the webas a source of information,be aware that there are many web-sites posted bywell-meaning individuals,which are full of wrong information.Nevertheless,browsing the web is a good exercise,since nothing is more important than tolearn to recognize which information is incorrect!Problem solving is very important in physics,and in order to obtain a work-ing knowledge of thermodynamics it is important to be able to do problems.Many problems are included,most of them with solutions.It is good to startproblems in class,and to have a discussion of the general approach that needsto be taken.When solving problems,for most people it is very beneﬁcial towork in groups,and that is encouraged.When you try to solve a problem andyou get stuck,do not look at the solution!Go to other textbooks and try toﬁnd material that pertains to your problem.When you believe that you havefound the solution,then it is time to compare with the solution in the notes,and then you can check if the solution in the notes is correct.In many cases,solving problems in thermodynamics always follows the samegeneral path.First you identify the independent state variables.If an exper-iment is performed at constant temperature,temperature is an independentstate variable because it is controlled.Control means that either we can set itat a certain value,or that we can prevent changes in the variable.For example,if we discuss a gas in a closed container,the volume of the gas is an independentstate variable,since the presence of the container makes it impossible for theXII INTRODUCTIONgas to expand or contract.Pressure,on the other hand,is not an independentstate variable in this example,since we have no means of controlling it.Second,based on our determination of independent state variables,we select the cor-rect thermodynamic potential to work with.Finally,we calculate the responsefunctions using this potential,and ﬁnd relations between these functions.Orwe use these response functions to construct equations of state using measureddata.And so on.Problem solving is,however,only a part of learning.Another part is to askquestions.Why do I think this material is introduced at this point?What isthe relevance?How does it build on the previous material?Sometimes thesequestions are subjective,because what is obvious for one person can be obscurefor another.The detailed order of topics might work well for one person but notfor another.Consequently,it is also important to ask questions about your ownlearning.How did I understand the material?Which steps did I make?Whichsteps were in the notes,and which were not?How did I ﬁll in the gaps?Insummary,one could say that problem solving improves technical skills,whichleads to a better preparation to apply the knowledge.Asking questions improvesconceptual knowledge,and leads to a better understanding how to approach newsituations.Asking questions about learning improves the learning process itselfand will facilitate future learning,and also to the limits of the current subject.Work in progress is adding more questions in the main text.There aremore in the beginning than in the end,a common phenomenon.As part oftheir assignments,I asked students in the beginning which questions they wouldintroduce.These questions are collected in an appendix.So,one should nottake these questions as questions from the students (although quite a few are),but also as questions that the students think are good to ask!In addition,Iasked students to give a summary of each chapter.These responses are alsogiven in an appendix.I provided this material in the appendices,because I think it is useful intwo diﬀerent ways.If you are a student studying thermodynamics,it is goodto know what others at your level in the educational system think.If you arestruggling with a concept,it is reassuring to see that others are too,and tosee with what kind of questions they came up to ﬁnd a way out.In a similarmanner,it it helpful to see what others picked out as the most important partof each chapter.By providing the summaries I do not say that I agree withthem (in fact,sometimes I do not)(Dutchmen rarely agree anyhow),but it givesa standard for what others picked out as important.And on the other hand,if you are teaching this course,seeing what students perceive to be the mostimportant part of the content is extremely helpful.Finally,a thanks to all students who took my classes.Your input has beenessential,your questions have lead to a better understanding of the material,and your research interests made me include a number of topics in these noteswhich otherwise would have been left out.History of these notes:1991Original notes for ﬁrst three chapters written using the program EXP.INTRODUCTION XIII1992Extra notes developed for chapters four and ﬁve.2001Notes on chapters four and ﬁve expanded.2002Notes converted to LATEX,signiﬁcantly updated,and chapter six added.2003Notes corrected,minor additions to ﬁrst ﬁve chapters,some additions tochapter six.2006Correction of errors.Updated section on ﬂuctuations.Added material oncorrelation functions in chapter seven,but this far from complete.2008Updated the material on correlation functions and included the two equa-tions of state related to pair correlation functions.2010Corrected errors and made small updates.1PART IThermodynamics Fundamentals2Chapter 1Basic Thermodynamics.The goal of this chapter is to introduce the basic concepts that we use in ther-modynamics.One might think that science has only exact deﬁnitions,but thatis certainly not true.In the history of any scientiﬁc topic one always startswith language.How do we describe things?Which words do we use and whatdo they mean?We need to get to some common understanding of what termsmean,before we can make them equivalent to some mathematical description.This seems rather vague,but since our natural way of communication is basedon language,it is the only way we can proceed.We all have some idea what volume means.But it takes some discussion todiscover that our ideas about volume are all similar.We are able to arrive atsome deﬁnitions we can agree on.Similarly,we all have a good intuition whattemperature is.More importantly,we can agree how we measure volume andtemperature.We can take an arbitrary object and put it in water.The rise ofthe water level will tell us what the volume of the object is.We can put anobject in contact with a mercury thermometer,and read of the temperature.We have used these procedures very often,since they are reproducible and giveus the same result for the same object.Actually,not the same,but in the sameGaussian distribution.We can do error analysis.In this chapter we describe the basic terms used in thermodynamics.Allthese terms are descriptions of what we can see.We also make connections withthe ideas of energy and work.The words use to do so are all familiar,and webuild on the vocabulary from a typical introductory physics course.We makemathematical connections between our newly deﬁned quantities,and postulatefour laws that hold for all systems.These laws are independent of the nature ofa system.The mathematical formulation by necessity uses many variables,andwe naturally connect with partial derivatives and multi-variable calculus.And then there is this quantity called temperature.We all have a good”feeling” for it,and standard measurements use physical phenomena like ther-mal expansion to measure it.We need to be more precise,however,and deﬁnetemperature in a complicated manner based on the eﬃciency of Carnot en-gines.At the end of the chapter we show that our deﬁnition is equivalent to34 CHAPTER 1.BASIC THERMODYNAMICS.the deﬁnition of temperature measured by an ideal gas thermometer.Once wehave made that connection,we know that our deﬁnition of temperature is thesame as the common one,since all thermometers are calibrated against idealgas thermometers.There are two reasons for us to deﬁne temperature in this complicated man-ner.First of all,it is a deﬁnition that uses energy in stead of a thermal materialsproperty as a basis.Second,it allows us to deﬁne an even more illustrious quan-tity,named entropy.This new quantity allows us to deﬁne thermal equilibriumin mathematical terms as a maximum of a function.The principle of maximumentropy is the corner stone for all that is discussed in the chapter that follow.1.1 Introduction.What state am I in?Simple beginnings.In the mechanical world of the 19th century,physics was very easy.All youneeded to know were the forces acting on particles.After that it was simply F =ma.Although this formalism is not hard,actual calculations are only feasiblewhen the system under consideration contains a few particles.For example,the motion of solid objects can be described this way if they are considered tobe point particles.In this context,we have all played with Lagrangians andHamiltonians.Liquids and gases,however,are harder to deal with,and areoften described in a continuum approximation.Everywhere in space one deﬁnesa mass density and a velocity ﬁeld.The continuity and Newton’s equations thenlead to the time evolution of the ﬂow in the liquid.Asking the right questions.The big diﬀerence between a solid and a liquid is complexity.In ﬁrst approx-imation a solid can be described by six coordinates (center of mass,orientation),while a liquid needs a mass density ﬁeld which is essentially an inﬁnite set ofcoordinates.The calculation of the motion of a solid is relatively easy,especiallyif one uses conservation laws for energy,momentum,and angular momentum.The calculation of the ﬂow of liquids is still hard,even today,and is often doneon computers using ﬁnite diﬀerence or ﬁnite element methods.In the 19th cen-tury,only the simplest ﬂow patterns could be analyzed.In many cases theseﬂow patterns are only details of the overall behavior of a liquid.Very often oneis interested in more general quantities describing liquids and gases.In the 19thcentury many important questions have been raised in connection with liquids1.1.INTRODUCTION.5and gases,in the context of steam engines.How eﬃcient can a steam enginebe?Which temperature should it operate at?Hence the problem is what weneed to know about liquids and gases to be able to answer these fundamentalquestions.Divide and conquer.In thermodynamics we consider macroscopicsystems,or systems with a largenumber of degrees of freedom.Liquids and gases certainly belong to this class ofsystems,even if one does not believe in an atomic model!The only requirementis that the system needs a description in terms of density and velocity ﬁelds.Solids can be described this way.In this case the mass density ﬁeld is givenby a constant plus a small variation.The time evolution of these deviationsfrom equilibrium shows oscillatory patterns,as expected.The big diﬀerencebetween a solid and a ﬂuid is that the deviations from average in a solid aresmall and can be ignored in ﬁrst approximation.No details,please.In a thermodynamic theory we are never interested in the detailed functionalform of the density as a function of position,but only in macroscopic or globalaverages.Typical quantities of interest are the volume,magnetic moment,andelectric dipole moment of a system.These macroscopic quantities,which canbe measured,are called thermodynamicor statevariables.They uniquely de-termine the thermodynamic stateof a system.State of a system.The deﬁnition of the state of a system is in terms of operations.Whatare the possible quantities which we can measure?In other words,how do weassign numbers to the ﬁelds describing a material?Are there any problems?For example,one might think that it is easy to measure the volume of a liquidin a beaker.But how does this work close to the critical point where the indexof refraction of the liquid and vapor become the same?How does this work ifthere is no gravity?In thermodynamics we simply assume that we are able tomeasure some basic quantities like volume.Another question is how many state variables do we need to describe asystem.For example we prepare a certain amount of water and pour it in abeaker,which we cap.The next morning we ﬁnd that the water is frozen.Itis obvious that the water is not in the same state,and that the information wehad about the systemwas insuﬃcient to uniquely deﬁne the state of the system.In this case we omitted temperature.It can be more complicated,though.Suppose we have prepared many ironbars always at the same shape,volume,mass,and temperature.They all look6 CHAPTER 1.BASIC THERMODYNAMICS.the same.But then we notice that some bars aﬀect a compass needle while othersdo not.Hence not all bars are the same and we need additional information touniquely deﬁne the state of the bar.Using the compass needle we can measurethe magnetic ﬁeld and hence we ﬁnd the magnetic moment of the bar.In a next step we use all bars with a given magnetic moment.We apply aforce on these bars and see how much they expand,from which we calculate theelastic constants of the bar.We ﬁnd that diﬀerent bars have diﬀerent values ofthe elastic constants.What are we missing in terms of state variables?This iscertainly not obvious.We need information about defects in the structure,orinformation about the mass density ﬁeld beyond the average value.Was I in a diﬀerent state before?Is it changing?At this point it is important to realize that a measured value of a statevariable is always a time-average.The pressure in an ideal gas ﬂuctuates,butthe time scale of the ﬂuctuations is much smaller than the time scale of theexternally applied pressure changes.Hence these short time ﬂuctuations can beignored and are averaged out in a measurement.This does imply a warning,though.If the ﬂuctuations in the state variables are on a time scale comparablewith the duration of the experiment a standard thermodynamic descriptionis useless.If they are on a very long time scale,however,we can use ourthermodynamic description again.In that case the motion is so slow and wecan use linear approximations for all changes.How do we change?The values of the state variables for a given system can be modiﬁed by ap-plying forces.An increase in pressure will decrease the volume,a change inmagnetic induction will alter the magnetic moment.The pressure in a gas ina container is in many cases equal to the pressure that this container exerts onthe gas in order to keep it within the volume of the container.It is possible touse this pressure to describe the state of the system and hence pressure (andmagnetic induction) are also state variables.One basic question in thermody-namics is how these state variables change when external forces are applied.Ina more general way,if a speciﬁc state variable is changed by external means,how do the other state variables respond?Number of variables,again.1.2.SOME DEFINITIONS.7The number of state variables we need to describe the state of a system de-pends on the nature of that system.We expand somewhat more on the previousdiscussion.An ideal gas,for example,is in general completely characterized byits volume,pressure,and temperature.It is always possible to add more statevariables to this list.Perhaps one decides to measure the magnetic moment ofan ideal gas too.Obviously,that changes our knowledge of the state of the idealgas.If the value of this additional state variable is always the same,no matterwhat we do in our experiment,then this variable is not essential.But one canalways design experiments in which this state variable becomes essential.Themagnetic moment is usually measured by applying a very small magnetic in-duction to the system.This external ﬁeld should be zero for all purposes.If itis not,then we have to add the magnetic moment to our list of state variables.It is also possible that one is not aware of additional essential state variables.Experiments will often indicate that more variables are needed.An example isan experiment in which we measure the properties of a piece of iron as a functionof volume,pressure,and temperature.At a temperature of about 770◦C someabnormal behavior is observed.As it turns out,iron is magnetic below thistemperature and in order to describe the state of an iron sample one has toinclude the magnetic moment in the list of essential state variables.An idealgas in a closed container is a simple system,but if the gas is allowed to escapevia a valve,the number of particles in this gas also becomes an essential statevariable needed to describe the state of the system inside the container.Are measured values always spatial averages?Are there further classiﬁcations of states or processes?1.2 Some deﬁnitions.Two types of processes.If one takes a block of wood,and splits it into two pieces,one has performeda simple action.On the level of thermodynamic variables one writes somethinglike V = V1+V2for the volumes and similar equations for other state variables.The detailed nature of this process is,however,not accessible in this language.In addition,if we put the two pieces back together again,they do in general notstay together.The process was irreversible.In general,in thermodynamics oneonly studies the reversiblebehavior of macroscopic systems.An example would8 CHAPTER 1.BASIC THERMODYNAMICS.be the study of the liquid to vapor transition.Material is slowly transportedfrom one phase to another and can go back if the causes are reversed.Thestate variables one needs to consider in this case are the pressure,temperature,volume,interface area (because of surface tension),and perhaps others in morecomplicated situations.When there is NO change.Obviously,all macroscopic systems change as a function of time.Most ofthese changes,however,are on a microscopic level and are not of interest.Weare not able to measure themdirectly.Therefore,in thermodynamics one deﬁnesa steady statewhen all thermodynamic variables are independent of time.Aresistor connected to a constant voltage is in a steady state.The current throughthe resistor is constant and although there is a ﬂow of charge,there are no netchanges in the resistor.The same amount of charge comes in as goes out.Thermodynamic equilibriumdescribes a more restricted situation.A systemis in thermodynamic equilibriumif it is in a steady state and if there are nonet macroscopic currents (of energy,particles,etc) over macroscopic distances.There is some ambiguity in this deﬁnition,connected to the scale and magnitudeof the currents.A vapor-liquid interface like the ocean,with large waves,isclearly not in equilibrium.But how small do the waves have to be in order thatwe can say that the system is in equilibrium?If we discuss the thermal balancebetween oceans and atmosphere,are waves important?Also,the macroscopiccurrents might be very small.Glass,for example,is not in thermal equilibriumaccording to a strict deﬁnition,but the changes are very slow with a time scaleof hundreds of years.Hence even if we cannot measure macroscopic currents,they might be there.We will in general ignore these situations,since they tendnot to be of interest on the time scale of the experiments!What do you think about hysteresis loops in magnets?State functions.Once we understand the nature of thermal equilibrium,we can generalizethe concept of state variables.A state functionis any quantity which in thermo-dynamic equilibrium only depends on the values of the state variables and noton the history (or future?) of the sample.A simple state function would be theproduct of the pressure and volume.This product has a physical interpretation,but cannot be measured directly.Two types of variables.1.2.SOME DEFINITIONS.9Thermodynamic variables come in two varieties.If one takes a system inequilibrium the volume of the left half is only half the total volume (surprise)but the pressure in the left half is equal to the pressure of the total system.There are only two possibilities.Either a state variable scales linearly with thesize of the system,or is independent of the size of the system.In other words,if we consider two systems in thermal equilibrium,made of identical material,one with volume V1and one with volume V2,a state variable X either obeysX1V1=X2V2or X1= X2.In the ﬁrst case the state variable is called extensiveandin the second case it is called intensive.Extensive state variables correspond togeneralized displacements.For the volume this is easy to understand;increasingvolume means displacing outside material and doing work on it in the process.Intensive state variables correspond to generalized forces.The pressure is theforce needed to change the volume.For each extensive state variable there is acorresponding intensive state variable and vice-versa.Extensive state variables correspond to quantities which can be determined,measured,or prescribed directly.The volume of a gas can be found by measur-ing the size of the container,the amount of material can be measured using abalance.Intensive state variables are measured by making contact with some-thing else.We measure temperature by using a thermometer,and pressure usinga manometer.Such measurements require equilibrium between the sample andmeasuring device.Note that this distinction limits where and how we can apply thermodynam-ics.The gravitational energy of a large systemis not proportional to the amountof material,but to the amount of material to the ﬁve-thirds power.If the forceof gravity is the dominant force internally in our systemwe need other theo-ries.Electrical forces are also of a long range,but because we have both positiveand negative charges,they are screened.Hence in materials physics we normallyhave no fundamental problems with applying thermodynamics.Thermodynamic limit.At this point we are able to deﬁne what we mean by a large system.Ratiosof an extensive state variable and the volume,likeXV,are often called densities.It is customary to write these densities in lower case,x =XV.If the volume is toosmall,x will depend on the volume.Since X is extensive,this is not supposed tobe the case.In order to get rid of the eﬀects of a ﬁnite volume (surface eﬀects!)one has to take the limit V → ∞.This is called the thermodynamic limit.All our mathematical formulas are strictly speaking only correct in this limit.In practice,this means that the volume has to be large enough in order thatchanges in the volume do not change the densities anymore.It is always possibleto write x(V ) = x∞+αV−1+O(V−2).The magnitude of α decides which valueof the volume is large enough.Physics determines the relation between state variables.10 CHAPTER 1.BASIC THERMODYNAMICS.Why are all these deﬁnitions important?So far we have not discussed anyphysics.If all the state variables would be independent we could stop right here.Fortunately,they are not.Some state variables are related by equations of stateand these equations contain the physics of the system.It is important to notethat these equations of state only relate the values of the state variables whenthe system is in thermal equilibrium,in the thermodynamic limit!If a systemis not in equilibrium,any combination of state variables is possible.It is evenpossible to construct non-equilibrium systems in which the actual deﬁnition ormeasurement of certain state variables is not useful or becomes ambiguous.Simple examples of equations of state are the ideal gas law pV = NRT andCurie’s law M =CNHT.The ﬁrst equation relates the product of the pres-sure p and volume V of an ideal gas to the number of moles of gas N and thetemperature T.The constant of proportionality,R,is the molar gas constant,which is the product of Boltzmann’s constant kBand Avogadro’s number NA.The second equation relates the magnetic moment M to the number of molesof atoms N,the magnetic ﬁeld H,and the temperature T.The constant of pro-portionality is Curie’s constant C.Note that in thermodynamics the preferredway of measuring the amount of material is in terms of moles,which again canbe deﬁned independent of a molecular model.Note,too,that in electricity andmagnetism we always use the magnetization density in the Maxwell equations,but that in thermodynamics we deﬁne the total magnetization as the relevantquantity.This makes M an extensive quantity.Equations of state are related to state functions.For any system we candeﬁne the state function Fstate= pV −NRT.It will take on all kinds of values.We then deﬁne the special class of systems for which Fstate≡ 0,identical tozero,as an ideal gas.The right hand side then leads to an equation of state,which can be used to calculate one of the basic state variables if others areknown.The practical application of this idea is to look for systems for whichFstateis small,with small deﬁned in an appropriate context.In that case we canuse the class with zero state function as a ﬁrst approximation of the real system.In many cases the ideal gas approximation is a good start for a description of areal gas,and is a start for systematic improvements of the description!How do we get equations of state?Equations of state have two origins.One can completely ignore the micro-scopic nature of matter and simply postulate some relation.One then uses thelaws of thermodynamics to derive functional forms for speciﬁc state variables asa function of the others,and compares the predicted results with experiment.The ideal gas law has this origin.This procedure is exactly what is done in ther-modynamics.One does not need a model for the detailed nature of the systems,but derives general conclusions based on the average macroscopic behavior of asystem in the thermodynamic limit.In order to derive equations of state,however,one has to consider the micro-scopic aspects of a system.Our present belief is that all systems consist of atoms.1.2.SOME DEFINITIONS.11If we know the forces between the atoms,the theory of statistical mechanics willtell us how to derive equations of state.There is again a choice here.It is pos-sible to postulate the forces.The equations of state could then be derived frommolecular dynamics calculations,for example.The other route derives theseeﬀective forces from the laws of quantum mechanics and the structure of theatoms in terms of electrons and nuclei.The interactions between the particlesin the atoms are simple Coulomb interactions in most cases.These Coulombinteractions follow fromyet a deeper theory,quantumelectro-dynamics,and areonly a ﬁrst approximation.These corrections are almost always unimportant inthe study of materials and only show up at higher energies in nuclear physicsexperiments.Why do we need equations of state?Equations of states can be used to classify materials.They can be usedto derive atomic properties of materials.For example,at low densities a gasof helium atoms and a gas of methane atoms both follow the ideal gas law.This indicates that in this limit the internal structure of the molecules does notaﬀect the motion of the molecules!In both cases they seem to behave like pointparticles.Later we will see that other quantities are diﬀerent.For example,theinternal energy certainly is larger for methane where rotations and translationsplay a role.Classification of changes of state.Since a static universe is not very interesting,one has to consider changesin the state variables.In a thermodynamic transformation or processa systemchanges one or more of its state variables.A spontaneousprocess takes placewithout any change in the externally imposed constraints.The word constraintin this context means an external description of the state variables for the sys-tem.For example,we can keep the volume of a gas the same,as well as thetemperature and the amount of gas.Or if the temperature of the gas is higherthan the temperature of the surroundings,we allow the gas to cool down.Inan adiabaticprocess no heat is exchanged between the system and the environ-ment.A process is called isothermalif the temperature of the system remainsthe same,isobaricif the pressure does not change,and isochoricif the massdensity (the number of moles of particles divided by the volume) is constant.Ifthe change in the system is inﬁnitesimally slow,the process is quasistatic.Reversible process.The most important class of processes are those in which the system startsin equilibrium,the process is quasistatic,and all the intermediate states and theﬁnal state are in equilibrium.These processes are called reversible.The process12 CHAPTER 1.BASIC THERMODYNAMICS.can be described by a continuous path in the space of the state variables,andthis path is restricted to the surfaces determined by the equations of state forthe system.By inverting all external forces,the path in the space of the statefunctions will be reversed,which prompted the name for this type of process.Reversible processes are important because they can be described mathemati-cally via the equations of state.This property is lost for an irreversible processbetween two equilibrium states,where we only have a useful mathematical de-scription of the initial and ﬁnal state.As we will see later,the second law ofthermodynamics makes another distinction between reversible and irreversibleprocesses.How does a process become irreversible?An irreversible process is either a process which happens too fast or which isdiscontinuous.The sudden opening of a valve is an example of the last case.Thesystem starts out in equilibrium with volume Viand ends in equilibrium witha larger volume Vf.For the intermediate states the volume is not well deﬁned,though.Such a process takes us outside of the space of state variables weconsider.It can still be described in the phase space of all system variables,andmathematically it is possible to deﬁne the volume,but details of this deﬁnitionwill play a role in the description of the process.Another type of irreversibleprocess is the same expansion from Vito Vfin a controlled way.The volumeis well-deﬁned everywhere in the process,but the system is not in equilibriumin the intermediate states.The process is going too fast.In an ideal gas thiswould mean,for example,pV ̸= NRT for the intermediate stages.Are there general principles connecting the values of state variables,valid forall systems?1.3 Zeroth Law of Thermodynamics.General relations.An equation of state speciﬁes a relation between state variables which holdsfor a certain class of systems.It represents the physics particular to that system.There are,however,a few relations that hold for all systems,independent of thenature of the system.Following an old tradition,these relations are called thelaws of thermodynamics.There are four of them,numbered 0 through 3.Themiddle two are the most important,and they have been paraphrased in thefollowing way.Law one tells you that in the game of thermodynamics youcannot win.The second law makes it even worse,you cannot break even.1.4.FIRST LAW:ENERGY.13Law zero.The zeroth law is relatively trivial.It discusses systems in equilibrium.Two systems are in thermal equilibrium if they are in contact and the totalsystem,encompassing the two systems as subsystems,is in equilibrium.In otherwords,two systems in contact are in equilibrium if the individual systems are inequilibriumand there are no net macroscopic currents between the systems.Thezeroth law states that if equilibrium system A is in contact and in equilibriumwith systems B and C (not necessarily at the same time,but Adoes not change),then systems B and C are also in equilibriumwith each other.If B and C are notin contact,it would mean that if we bring them in contact no net macroscopiccurrents will ﬂow.Significance of law zero.The importance of this law is that it enables to deﬁne universal standardsfor temperature,pressure,etc.If two diﬀerent systems cause the same readingon the same thermometer,they have the same temperature.A temperaturescale on a new thermometer can be set by comparing it with systems of knowntemperature.Therefore,the ﬁrst law is essential,without it we would not beable to give a meaningful analysis of any experiment.We postulate it a a law,because we have not seen any exceptions.We postulate it as a law,because weabsolutely need it.We cannot prove it to be true.It cannot be derived fromstatistical mechanics,because in that theory it is also a basic or fundamentalassumption.But if one rejects it completely,one throws away a few hundredyears of successful science.But what if the situation is similar to Newton’sF = ma,where Einstein showed the limits of validity?That scenario is certainlypossible,but we have not yet needed it,or seen any reason for its need.Also,it is completely unclear what kind of theory should be used to replace all whatwe will explore in these notes.Are there any consequences for the sizes of the systems?1.4 First law:Energy.Heat is energy flow.The ﬁrst law of thermodynamics states that energy is conserved.The changein internal energy U of a system is equal to the amount of heat energy added tothe system minus the amount of work done by the system.It implies that heat14 CHAPTER 1.BASIC THERMODYNAMICS.is a form of energy.Technically,heat describes the ﬂow of energy,but we arevery sloppy in our use of words here.The formal statement of the ﬁrst law isdU =¯dQ−¯dW (1.1)The amount of heat added to the systemis¯dQ and the amount of work doneby the system is¯dW.The mathematical formulation of the ﬁrst law also showsan important characteristic of thermodynamics.It is often possible to deﬁnethermodynamic relations only via changes in the thermodynamic quantities.Note that we deﬁne the work term as work done on the outside world.Itrepresents a loss of energy of the system.This is the standard deﬁnition,andrepresents the fact that in the original analysis one was interested in supplyingheat to an engine,which then did work.Some books,however,try to be consis-tent,and write the work term as work done on the system.In that case thereis no minus sign in equation 1.1.Although that is somewhat neater,it causestoo much confusion with standard texts.It is always much too easy to lose aminus sign.The ﬁrst law again is a statement that has always been observed to betrue.In addition,the ﬁrst law does follow directly in a statistical mechanicaltreatment.We have no reason to doubt the validity of the ﬁrst law,and wediscard any proposals of engines that create energy out of nothing.But again,there is no absolute proof of its validity.And,again as well,if one discardsthe ﬁrst law,all the remainder of these notes will be invalid as well.A theorywithout the ﬁrst law is very diﬃcult to imagine.The internal energy is a state variable.The internal energy U is a state variable and an inﬁnitesimal change ininternal energy is an exact diﬀerential.Since U is a state variable,the value ofany integral∫dU depends only on the values of U at the endpoints of the pathin the space of state variables,and not on the speciﬁc path between the end-points.The internal energy U has to be a state variable,or else we could devisea process in which a systemgoes through a cycle and returns to its original statewhile loosing or gaining energy.For example,this could mean that a burningpiece of coal today would produce less heat than tomorrow.If the internalenergy would not be a state variable,we would have sources of free energy.Exact differentials.The concept of exact diﬀerentialsis very important,and hence we will illus-trate it by using some examples.Assume the function f is a state function ofthe state variables x and y only,f(x,y).For small changes we can writedf =(∂f∂x)ydx +(∂f∂y)xdy (1.2)1.4.FIRST LAW:ENERGY.15where in the notation for the partial derivatives the variable which is kept con-stant is also indicated.This is always very useful in thermodynamics,becauseone often changes variables.There would be no problems if quantities were de-ﬁned directly like f(x,y) = x+y.In thermodynamics,however,most quantitiesare deﬁned by or measured via small changes in a system.Hence,suppose thechange in a quantity g is related to changes in the state variables x and y via¯dg = h(x,y)dx +k(x,y)dy (1.3)Is the quantity g a state function,in other words is g uniquely determinedby the state of a system or does it depend on the history,on how the systemgot into that state?It turns out that a necessary and suﬃcient condition for gto be a state function is that(∂h∂y)x=(∂k∂x)y(1.4)everywhere in the x-y space.The necessity follows immediately from 1.2,aslong as we assume that the partial derivatives in 1.4 exist and are continuous.This is because in second order derivatives we can interchange the order of thederivatives under such conditions.That it is suﬃcient can be shown as follows.Consider a path (x,y) = (ϕ(t),ψ(t)) from (x1,y1) at t1to (x2,y2) at t2andintegrate¯dg,using dx =dϕdtdt,dy =dψdtdt,∫t2t1(h(ϕ(t),ψ(t))dϕdt+k(ϕ(t),ψ(t))dψdt)dt (1.5)DeﬁneH(x,y) =∫x0dx′h(x′,y) +∫y0dy′k(0,y′) (1.6)and H(t) = H(ϕ(t),ψ(t)).It follows thatdHdt=(∂H∂x)ydϕdt+(∂H∂y)xdψdt(1.7)The partial derivatives of H are easy to calculate:(∂H∂x)y(x,y) = h(x,y) (1.8)(∂H∂y)x(x,y) =∫x0dx′(∂h∂y)x(x′,y) +k(0,y) =∫x0dx′(∂k∂x)y(x′,y) +k(0,y) = k(x,y) (1.9)This implies that16 CHAPTER 1.BASIC THERMODYNAMICS.∫t2t1¯dg =∫t2t1dHdt(1.10)and hence the integral of¯dg is equal to H(t2)−H(t1) which does not dependon the path taken between the end-points of the integration.Example.An example might illustrate this better.Suppose x and y are two state vari-ables,and they determine the internal energy completely.If we deﬁne changesin the internal energy via changes in the state variables x and y viadU = x2ydx +13x3dy (1.11)we see immediately that this deﬁnition is correct,the energy U is a state func-tion.The partial derivatives obey the symmetry relation 1.4 and one can simplyintegrate dU and check that we get U(x,y) =13x3y +U0.The changes in heat and work are now assumed to be related in the followingway¯dQ =12x2ydx +12x3dy (1.12)¯dW = −12x2ydx +16x3dy (1.13)These deﬁnitions do indeed obey the ﬁrst law 1.1.It is also clear using thesymmetry relation 1.4 that these two diﬀerentials are not exact.Suppose the systemwhich is described above is originally in the state (x,y) =(0,0).Now we change the state of the system by a continuous transformationfrom (0,0) to (1,1).We do this in two diﬀerent ways,however.Path one takesus from (0,0) to (0,1) to (1,1) along two straight line segments,path two issimilar from (0,0) to (1,0) to (1,1).The integrals for dU,¯dQ,and¯dW areeasy,since along each part of each path either dx or dy is zero.First take path one.U(1,1) −U(0,0) =∫10dy13(0)3+∫10dxx21 =13(1.14)∆Q =∫10dy12(0)3+∫10dx12x21 =16(1.15)∆W =∫10dy16(0)3+∫10dx(−12)x21 = −16(1.16)First of all,the change in U is consistent with the state function we found,U(x,y) =13x3y +U0.Second,we have ∆U = ∆Q−∆W indeed.It is easy to1.4.FIRST LAW:ENERGY.17calculate that for the second path we have ∆U =13,∆Q =12,and ∆W =16.The change in internal energy is indeed the same,and the ﬁrst law is againsatisﬁed.Importance of Q and W not being state functions.Life on earth would have been very diﬀerent if Q and W would have beenstate variables.Steam engines would not exist,and you can imagine all conse-quences of that fact.Expand on the consequences of Q and W being state functionsAny engine repeats a certain cycle over and over again.A complete cyclein our example above might be represented by a series of continuous changes inthe state variables (x,y) like (0,0) →(0,1) →(1,1) →(1,0) →(0,0).After thecompletion of one cycle,the energy U is the same as at the start of the cycle.The change in heat for this cycle is ∆Q =16−12= −13and the work done onthe environment is ∆W = −16−16= −13.This cycle represents a heater:since∆Q is negative,heat is added to the environment and since ∆W is negativethe environment does work on the system.Running the cycle in the oppositedirection yields an engine converting heat into work.If Q and Wwould be statevariables,for each complete cycle we would have ∆Q = ∆W = 0,and no netchange of work into heat and vice-versa would be possible!When was the ﬁrst steam engine constructed?Work can be done in many diﬀerent ways.A change in any of the extensivestate variables of the system will cause a change in energy,or needs a forcein order that it happens.Consider a system with volume V,surface area A,polarization⃗P,magnetic moment⃗M,and number of moles of material N.Thework done by the system on the environment is¯dW = pdV −σdA−⃗Ed⃗P −⃗Hd⃗M −µdN (1.17)where the forces are related to the intensive variables pressure p,surface tensionσ,electric ﬁeld⃗E,magnetic ﬁeld⃗H,and chemical potential µ.Note that sometextbooks treat the µdN term in a special way.There is,however,no formalneed to do so.The general form is¯dW = −∑jxjdXj(1.18)where the generalized force xjcauses a generalized displacement dXjin thestate variable Xj.18 CHAPTER 1.BASIC THERMODYNAMICS.The signs in work are normally negative.If we increase the total magneticmoment of a sample in an external magnetic ﬁeld,we have to add energy tothe sample.In other words,an increase in the total magnetic moment increasesthe internal energy,and work has to be done on the sample.The work doneby the sample is negative.Note that the pdV term has the opposite sign fromall others.If we increase the volume of a sample,we push outside materialaway,and do work on the outside.A positive pressure decreases the volume,while a positive magnetic ﬁeld increases the magnetic magnetization in general.This diﬀerence in sign is on one historical,and is justiﬁed by the old,intuitivedeﬁnitions of pressure and other quantities.But it also has a deeper meaning.Volume tells us how much space the sample occupies,while all other extensivequantities tell us how much of something is in that space.In terms of densities,volume is in the denominator,while all other variables are in the numerator.This gives a change in volume the opposite eﬀect from all other changes.1.5 Second law:Entropy.Clausius and Kelvin.The second law of thermodynamics tells us that life is not free.Accordingto the ﬁrst law we can change heat into work,apparently without limits.Thesecond law,however,puts restrictions on this exchange.There are two versionsof the second law,due to Kelvin and Clausius.Clausius stated that there areno thermodynamic processes in which the only net change is a transfer of heatfrom one reservoir to a second reservoir which has a higher temperature.Kelvinformulated it in a diﬀerent way:there are no thermodynamic processes in whichthe only eﬀect is to extract a certain amount of heat from a reservoir and convertit completely into work.The opposite is possible,though.These two statementsare equivalent as we will see.Heat is a special form of energy exchange.The second law singles out heat as compared to all other forms of energy.Since work is deﬁned via a change in the extensive state variables,we can thinkof heat as a change of the internal degrees of freedom of the system.Henceheat represents all the degrees of freedom we have swept under the rug whenwe limited state variables to measurable,average macroscopic quantities.Theonly reason that we can say anything at all about heat is that it is connected toan extremely large number of variables (because of the thermodynamic limit).In that case the mathematical laws of large numbers apply,and the statementsabout heat become purely statistical.In statistical mechanics we will returnto this point.Note that the second law does not limit the exchange of energyswitching from one form of work to another.In principal we could change1.5.SECOND LAW:ENTROPY.19mechanical work into electrical work without penalty!In practice,heat is alwaysgenerated.Heat as a measurable quantity.One important implicit assumption in these statements is that a large out-side world does exist.In the ﬁrst law we deﬁne the change of energy via anexchange of heat and work with the outside world.Hence we assume that thereis something outside our system.As a consequence,the second law does notapply to the universe as a whole.This is actually quite important.In thermo-dynamics we discuss samples,we observe sample,and we are on the outside.We have large reservoirs available to set pressure or temperature values.Hencewhen we take the thermodynamic limit for the sample,we ﬁrst have to take thelimit for the outside world and make it inﬁnitely large.This point will comeback in statistical mechanics,and the ﬁrst to draw attention to it was Maxwellwhen he deployed his demon.The deﬁnitions of heat and entropy in thermodynamics are based on quan-tities that we can measure.They are operational deﬁnitions.The second law isan experimental observation,which has never been falsiﬁed in macroscopic ex-periments.Maxwell started an important discussion trying to falsify the secondlaw on a microscopic basis (his famous demon),but that never worked either.It did lead to important statements about computing,though!If the second law is universally valid,it deﬁnes a preferred direction of time(by increasing entropy or energy stored in the unusable internal variables),andseems to imply that every systemwill die a heat death.This is not true,however,because we always invoke an outside world,and at some point heat will haveto ﬂow from the system to the outside.This is another interesting point ofdiscussion in the philosophy of science.In statistical mechanics we can deﬁne entropy and energy by consideringthe system only,and is seems possible to deﬁne the entropy of the universe inthat way.Here one has to keep in mind that the connection between statisticalmechanics and thermodynamics has to be made,and as soon as we make thatconnection we invoke an outside world.This is an interesting point of debate,too,which takes place on the same level as the debate in quantum mechanicsabout the interpretation of wave functions and changes in the wave functions.Carnot engine.We have seen before that machines that run in cycles are useful to do work.In the following we will consider such a machine.The important aspect of themachine is that every step is reversible.The second law leads to the importantconclusion that all reversible machines using the same process have the sameeﬃciency.We want to make a connection with temperature,and therefore wedeﬁne an engine with a cycle in which two parts are at constant temperatures,inorder to be able to compare values of these temperatures.The other two parts20 CHAPTER 1.BASIC THERMODYNAMICS.Figure 1.1:Carnot cycle in PV diagram.are simpliﬁed by making them adiabatic,so no heat is exchanged.Connectingthe workings of such engines with the second law will then allow us to deﬁne atemperature scale,and also deﬁne entropy.An engine is a system which changes its thermodynamic state in cycles andconverts heat into work by doing so.A Carnot engine is any system repeatingthe following reversiblecycle:(1) an isothermal expansion at a high tempera-ture T1,(2) an adiabatic expansion in which the temperature is lowered to T2,(3) an isothermal contraction at temperature T2,and ﬁnally (4) an adiabaticcontraction back to the initial state.In this case work is done using a changein volume.Similar Carnot engines can be deﬁned for all other types of work.Itis easiest to talk about Carnot engines using the pressure p,the volume V,andthe temperature T as variables.A diagram of a Carnot engine in the pV planeis shown in ﬁgure 1.1.The material in a Carnot engine can be anything.For practical reasons itis often a gas.Also,because steps one and three are isothermal,contact witha heat reservoir is required,and the Carnot engine operates between these twoheat reservoirs,by deﬁnition.Mechanical work is done in all four parts of thecycle.We can deﬁne Carnot engines for any type of work,but mechanical work isthe easiest to visualize (and construction of Carnot engines based on mechanicalwork is also most common).One also recognizes the historical importance ofsteam engines;such engines ”drove” the development of thermodynamics!Carnot engines are the most efficient!The second law of thermodynamics has a very important consequence forCarnot engines.One can show that a Carnot engine is the most eﬃcient engineoperating between two reservoirs at temperature T1and T2!This is a very strongstatement,based on minimal information.The eﬃciency η is the ratio of thework Wperformed on the outside world and the heat Q1absorbed by the system1.5.SECOND LAW:ENTROPY.21Figure 1.2:Schematics of a Carnot engine.in the isothermal step one at high temperature.Remember that in steps twoand four no heat is exchanged.The heat absorbed from the reservoir at lowtemperature in step three is Q2and the ﬁrst law tells us that W = Q1+Q2.We deﬁne the ﬂow of heat Qito be positivewhen heat ﬂows intothe system.In most engines we will,of course,have Q1> 0 and Q2< 0.This gives usη =WQ1= 1 +Q2Q1(1.19)Work is positive when it represents a ﬂow of energy to the outside world.ACarnot engine in reverse is a heater (or refrigerator depending on which reservoiryou look at).Can the eﬃciency be greater than one?A Carnot engine can be represented in as follows,see ﬁgure 1.2.In thisﬁgure the arrows point in the direction in which the energy ﬂow is deﬁned tobe positive.Equivalency of Clausius and Kelvin.The two formulations of the second lawof Clausius and Kelvin are equivalent.If a Kelvin engine existed which converts heat completely into work,this workcan be transformed into heat dumped into a reservoir at higher temperature,incontradiction with Clausius.If a Clausius process would exist,we can use it tostore energy at a higher temperature.A normal engine would take this amountof heat,dump heat at the low temperature again while performing work,and22 CHAPTER 1.BASIC THERMODYNAMICS.Figure 1.3:Two engines feeding eachother.there would be a contradiction with Kelvin’s formulation of the second law.Thestatement about Carnot engines is next shown to be true in a similar way.Contradictions if existence of more efficient engine.Assume that we have an engine X which is more eﬃcient than a Carnotengine C.We will use this engine X to drive a Carnot engine in reverse,seeﬁgure 1.3.The engine X takes an amount of heat QX> 0 from a reservoir athigh temperature.It produces an amount of work W = ηXQX> 0 and takesan amount of heat Q2X= (ηX−1)QXfrom the reservoir at low temperature.Notice that we need ηX< 1,(and hence Q2X< 0 ),otherwise we wouldviolate Kelvin’s formulation of the second law.This means that the net ﬂowof heat is towards the reservoir of low temperature.Now take a Carnot engineoperating between the same two reservoirs.This Carnot engine is driven bythe amount of work W,hence the amount of work performed by the Carnotengine is WC= −W.This Carnot engine takes an amount of heat Q1C=WCηC= −ηXηCQXfrom the reservoir at high temperature and an amount Q2C=WC− Q1C= (1ηC− 1)ηXQXfrom the reservoir at low temperature.Nowconsider the combination of these two engines.This is a machine which takesan amount of heat (1 −ηxηc)Qxfrom the reservoir at high temperature and theopposite amount from the reservoir at low temperature.Energy is conserved,but Clausius tells us that the amount of heat taken from the high temperaturereservoir should be positive,or ηx≤ ηc.Hence a Carnot engine is the mosteﬃcient engine which one can construct!In a diﬀerent proof we can combine an engine X and a Carnot engine,butrequire Q2X+ Q2C= 0.Such an engine produces an amount of work Wnetwhich has to be negative according to Kelvin.1.5.SECOND LAW:ENTROPY.23Show that this implies ηX≤ ηC.All Carnot engines are equally efficient.One can easily show that all Carnot engines have the same eﬃciency.Sup-pose the eﬃciencies of Carnot engine one and two are η1and η2,respectively.Use one Carnot engine to drive the other in reverse,and it follows that we needη1≤ η2and also η2≤ η1,or η1= η2.Hence the eﬃciency of an arbitraryCarnot engine is ηC.This is independent of the details of the Carnot engine,except that it should operate between a reservoir at T1and a reservoir at T2.These are the only two variables which play a role,and the Carnot eﬃciencyshould depend on them only:ηC(T1,T2).Carnot efficiency can be measured experimentally.This eﬃciency function can be determined experimentally by measuring Qand W ﬂowing in and out of a given Carnot engine.How?That is a problem.First,consider the work done.This is the easier part.For example,because ofthe work done a weight is lifted a certain distance.This gives us the changein energy,and hence the work done.In order to use this type of measurement,however,we need to know details about the type of work.This is essentiallythe same as saying that we need to understand the measurements we are doing.How do we measure heat?We need a reference.For example,take a largeclosed glass container with water and ice,initially in a one to one ratio.Assumethat the amount of energy to melt a unit mass of ice is our basic energy value.We can measure the amount of heat that went into this reference system bymeasuring the change in the volumes of water and ice.Also,if a sample ofunknown temperature is brought into contact with the reference system,we caneasily determine whether the temperature of the sample is higher or lower thenthe reference temperature of the water and ice system.If it is higher,ice willmelt,if it is lower,water will freeze.Note that we assume that the temperatureof the reference system is positive!Experimental definition of temperature.State variables are average,macroscopic quantities of a system which can bemeasured.This is certainly a good deﬁnition of variables like volume,pressure,and number of particles.They are related to basic concepts like length,mass,charge,and time.Temperature is a diﬀerent quantity,however.A practicaldeﬁnition of the temperature of an object is via a thermometer.The activesubstance in the thermometer could be mercury or some ideal gas.But thoseare deﬁnitions which already incorporate some physics,like the linear expansionof solids for mercury or the ideal gas law for a gas.It is diﬀerent from the24 CHAPTER 1.BASIC THERMODYNAMICS.deﬁnitions of length and time in terms of the standard meter and clock.In asimilar vein we would like to deﬁne temperature as the result of a measurementof a comparison with a standard.Hence we assume that we have a known objectof temperature T0,similar to the standard meter and clock.An example wouldbe the container with the water and ice mixture mentioned above.Now how do we compare temperatures on a quantitative level?If we wantto ﬁnd the temperature of an object of unknown temperature T,we take aCarnot engine and operate that engine between the object and the standard.Wemeasure the amount of heat Q ﬂowing from the reference system to the Carnotengine or from the Carnot engine to the reference system.We also measurethe amount of work W done by the Carnot engine.We use the ﬁrst law todetermine the amount of heat ﬂowing out of the high temperature reservoir,ifneeded.The ratio of these two quantities is the eﬃciency of the Carnot engine,which only depends on the two temperatures.We ﬁrst determine if the object has a higher or lower temperature thenthe reference by bringing them in direct contact.If ice melts,the object waswarmer,if ice forms,it was colder.If the temperature of the reference system is higher that the temperature ofthe object,we use the reference system as the high temperature reservoir.Wemeasure the amount of heat Q going out of the reference system and ﬁnd:ηC=WQ(1.20)If the temperature of the reference system is lower that the temperature ofthe object,we use the reference system as the low temperature reservoir.Wemeasure the amount of heat Q going out of the reference system,which is nownegative,since heat is actually going in,and ﬁnd:ηC=WW −Q(1.21)In the ﬁrst case we assign a temperature T to the object according toTT0= (1 −ηC) (1.22)and in the second case according toT0T= (1 −ηC) (1.23)This is our denition of temperature on the Carnot scale.It is an im-portant step forward,based on the unique eﬃciency of Carnot engines,which initself is based on the second law.Theoretically,this is a good deﬁnition becauseCarnot engines are well-deﬁned.Also,energy is well-deﬁned.The importantquestion,of course,is how this deﬁnition relates to known temperature scales.We will relate the Carnot temperature scale to the ideal gas temperature scaleat the end of this chapter.1.5.SECOND LAW:ENTROPY.25Figure 1.4:Two Carnot engines in series.Efficiency for arbitrary temperatures.We can analyze the general situation for a Carnot engine between arbitrarytemperatures as follows.Assume that we have T > T0> T′,all other caseswork similarly.Consider the following couple of Carnot engines (see ﬁgure 1.4) and demand that Q′1+Q2= 0 (no heat going in or out the reference system).Argue that this is equivalent to a single Carnot engine working between T andT′.For this system we haveT′T0= (1 − η′C) andT0T= (1 − ηC),orT′T= (1 −η′C)(1 −ηC).The eﬃciencies can be expressed in the energy exchanges and wehaveT′T= (1 −W′Q′1)(1 −WQ1).But we have Q′1= −Q2= Q1− W and henceT′T= (1 −W′Q1−W)(1 −WQ1).The right hand side is equal to 1 −WQ1−W′Q1−W(1 −WQ1) = 1 −WQ1−W′Q1.In other words:T′T= 1 −W +W′Q1= 1 −ηC(1.24)where the relation now holds for arbitrary values of the temperature.Can we obtain negative values of the temperature?Carnot cycle again.26 CHAPTER 1.BASIC THERMODYNAMICS.Using the temperature scale deﬁned by the Carnot engine,we can reanalyzethe Carnot cycle.The eﬃciency is related to the heat ∆Q1absorbed in the ﬁrststep and ∆Q2absorbed in the third step (which is negative in an engine) byηC= 1 +∆Q2∆Q1(1.25)where we have used a notation with ∆Q to emphasize the fact that we lookat changes.But that is not really essential.Fromthe previous equation we ﬁnd:∆Q1T1+∆Q2T2= 0 (1.26)Since there is no heat exchanged in steps two and four of the Carnot cycle,thisis equivalent toIC¯dQT= 0 (1.27)where the closed contour C speciﬁes the path of integration in the space of statevariables.Integral for arbitrary cycles.Next we consider the combined eﬀect of two Carnot engines,one workingbetween T1and T2,the other one between T2and T3.Now compare this with asingle system which follows the thermodynamic transformation deﬁned by theoutside of the sumof the two Carnot contours.One can think of the total processas the sum of the two Carnot steps,introducing an intermediate reservoir,inwhich no net heat is deposited.The contour integral of¯dQTis also zero for thesingle process,since the two contributions over the common line are oppositeand cancel.Any general closed path in the space of state variables,restricted tothose surfaces which are allowed by the equations of state,can be approximatedas the sum of a number of Carnot cycles with temperature diﬀerence ∆T.Theerror in this approximation approaches zero for ∆T →0.Hence:IR¯dQT= 0 (1.28)where R is an arbitrary cyclic,reversible process.Definition of entropy.Formula 1.28 has the important consequence that2∫1¯dQTis path independent.We deﬁne a new variable S byS2= S1+∫21¯dQT(1.29)1.5.SECOND LAW:ENTROPY.27and because the integral is path independent S is a state function.When theintegration points are close together we getTdS =¯dQ (1.30)in which dS is an exact diﬀerential.The quantity S is called the entropy.Inthermodynamics we deﬁne the entropy froma purely macroscopic point of view.It is related to inﬁnitesimally small exchanges of thermal energy by requiringthat the diﬀerential¯dQ can be transformed into an exact diﬀerential by multi-plying it with a function of the temperature alone.One can always transform adiﬀerential into an exact diﬀerential by multiplying it with a function of all statevariables.In fact,there are an inﬁnite number of ways to do this.The restric-tion that the multiplying factor only depends on temperature uniquely deﬁnesthis factor,apart from a constant factor.One could also deﬁne 5TdS =¯dQ,which would simply re-scale all temperature values by a factor ﬁve.First law in exact differentials.The ﬁrst law of thermodynamics in terms of changes in the entropy isdU = TdS −¯dW (1.31)For example,if we consider a systemwhere the only interactions with the outsideworld are a possible exchange of heat and mechanical work,changes in theinternal energy are related to changes in the entropy and volume throughdU = TdS −pdV (1.32)An important note at this point is that we often use the equation 1.32 as amodel.It does indeed exemplify some basic concepts,but for real applicationsit is too simple.The simplest form of the ﬁrst law that has physical meaning isthe following:dU = TdS −pdV +µdN (1.33)Entropy is extensive.In the deﬁnition of the Carnot temperature of an object the size of the objectdoes not play a role,only the fact that the object is in thermal equilibrium.Asa consequence the temperature is an intensive quantity.On the other hand,ifwe compare the heat absorbed by a system during a thermodynamic processwith the heat absorbed by a similar system which is α times larger,it is nothard to argue that the amount of heat exchanged is α times larger as well.Asa consequence,the entropy S is an extensive state variable.28 CHAPTER 1.BASIC THERMODYNAMICS.Natural variables for the internal energy are all extensive statevariables.Changes in the internal energy U are related to changes in the extensivestate variables only,since the amount of work done is determined by changesin extensive state variables only and S is extensive.In this sense,the naturalset of variables for the state function U is the set of all extensive variables.Bynatural we mean the set of variables that show as diﬀerentials in the ﬁrst law,hence small changes are directly related.Importance of thermodynamic limit.Equation 1.31 has an interesting consequence.Suppose that we decide thatanother extensive state variable is needed to describe the state of a system.Hence we are adding a termxdX to¯dW.This means that the number of internaldegrees of freedom is reduced by one,since we are specifying one additionalcombination of degrees of freedom via X.This in its turn indicates that theentropy should change,since it is a representation of the internal degrees offreedom of the system.The deﬁnition of the entropy would therefore dependon the deﬁnition of work,which is an unacceptable situation.Fortunately,thethermodynamic limit comes to rescue here.Only when the number of degrees offreedom is inﬁnitely large,the change by one will not alter the entropy.Hencethe entropy is only well-deﬁned in the thermodynamic limit.Independent and dependent variables.From equation 1.32 we ﬁnd immediately thatT =(∂U∂S)V(1.34)andp = −(∂U∂V)S(1.35)which shows that in the set of variables p,V,T,S only two are independent.Ifwe know the basic physics of the system,we know the state function U(S,V ) andcan derive the values for p and T according to the two state functions deﬁned bythe partial derivatives.Functions of the form T = f(S,V ) and p = g(S,V ) arecalled equations of state.More useful forms eliminate the entropy from theseequations and lead to equations of state of the form p = h(T,V ).The relationU = u(S,V ) is called an energy equationand is not an equation of state,sinceit deﬁnes an energy as a function of the independent state variables.Suchrelations are the basis for equations of state,but we use equation of state onlywhen we describe state variables that occur in pairs,like T and S,or p and V.1.5.SECOND LAW:ENTROPY.29Equations of state give dependent state variables of this nature as a function ofindependent ones.In the next chapter we will discuss how to change variables and make com-binations like T and V independent and the others dependent.Entropy in terms of work.We now return to the deﬁnition of entropy according to equation 1.29 above.If we apply the ﬁrst law we getS2= S1+∫21dU +¯dWT(1.36)and if we express work in its generalized form according to 1.18 we seeS2= S1+∫21dU +∑jxjdXjT(1.37)which shows that entropy is deﬁned based on basic properties of the system,which can be directly measured.There is no device that measures entropydirectly,and in that sense it is diﬀerent from all other state variables.Butit is possible to perform processes in which the entropy does not change,andhence we do have control over changes in the entropy.Since entropy is basedon changes in other state variables and the internal energy,it is well deﬁnedin reversible processes.The second law singles out heat as a more restrictedchange of energy,and this has consequences for entropy.We now discuss thoseconsequences.Change in entropy in an irreversible process.Up to this point we only considered the entropy in connection with reversibleprocesses.Diﬀerent rules follow for irreversible processes.Consider a generalprocess in which heat is transferred from a reservoir at high temperature to areservoir at low temperature (and hence Q1> 0).The eﬃciency of this processis at most equal to the Carnot eﬃciency,and henceWQ1= 1 +Q2Q1≤ ηC= 1 −T2T1(1.38)For such a general process we haveQ